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nLab > Latest Changes: plus construction on presheaves

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Stephan Spahn added stuff to plus construction on presheaves

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Thanks, Stephan!

    you ask

    To address the question of compact objects in Sh ∞(SmoothMfd) there should be an (∞,1)-plus construction, too. Is in this case where the (∞,1)-site is just a 1-site somehow clear how this works?

    For nn-truncated objects it is in principle clear: one has to apply the plus-construction (n+2)-times in a row!

    See for instance section 6.5.3 of HTT.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    I have added a brief remark on this in the Idea section. But am out of time now.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2012
    • (edited Apr 30th 2012)

    I have added the reference to the famous Heller-Rowe article where the plus construction in the case of abelian sheaves is studied. In the abelian context one usually says Heller-Rowe functor. Interestingly, the theory of Q-categories of Rosenberg has been written out in 1988 to exhibit the common and nontrivial generalization of the functor H ℱH_{\mathcal{F}} in the theory of Gabriel localization and Heller/Rowe functor, as the instances of the same construction. Gabriel localization is also H ℱ 2H^2_{\mathcal{F}}, just like sheafification. Here ℱ\mathcal{F} is a Gabriel filter.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeApr 30th 2012

    I always thought that the fact that we needed to do (−) +(-)^+ twice had something to do with the fact that the equaliser diagram has two stages, but I never did find a good technical explanation of this point. What is clear, though, is that doing it once is a generalisation of computing Hˇ 0\check{H}^0. I imagine these two facts are related via nn-categories…

  6. Added to plus construction on presheaves that applying the plus construction once is not left adjoint to the inclusion of separated presheaves in all presheaves.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2021

    I just realized that the description of the plus-construction in 6.2.2.9 of HTT is exactly an instance of (the ∞\infty-version of) the description of a p.r.a. functor in terms of a polynomial of discrete fibrations here.

    That’s interesting.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2021

    Well, not quite. The last functor ρ\rho is not a discrete fibration. But it’s closely related.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2021

    The link to descent morphism didn’t match the usage of that term on that page.

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    The first and third definitions were really the same, so I collapsed them, and added in the internal description to the same list for good measure.

    diff, v13, current

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Added a definition in terms of Kan extensions, inspired by Lurie’s. Also remarked on which definitions work for ∞\infty-presheaves.

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Sketched a proof that the plus-construction is a well-pointed endofunctor whose fixedpoints are the sheaves, using an observation that there is a situation of cohesion involved.

    diff, v13, current

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Sketched a proof that the plus-construction is a well-pointed endofunctor whose fixedpoints are the sheaves, using an observation that there is a situation of cohesion involved.

    diff, v13, current

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Completed a proof, inspired by Anel and Leena Subramaniam, that the plus-construction for nn-separated presheaves converges after n+2n+2 steps.

    diff, v13, current

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2021

    Added link to the Coq formalization.

    diff, v16, current

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2021

    Added remarks about generalization to lex modulators and non-presheaf toposes.

    diff, v16, current

1 to 16 of 16

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